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strained from rotation. The local tangential strain was
found to be insignificant compared to the global radial
strain (Al-Aboodi 2006). This is due to the fact that the
roller is sliding inside the tube as in Williams (1997) and
Metzger et al. (1995). This reduces the amount of friction
and tangential strain.
Similar to the work of Williams (1997) four rolling
revolutions and two smoothing revolutions were used. No radial displacement is assigned during the smoothing pro-
cess. To take into account the non-linear response of tube
material in the elastic–plastic transition region and in the
plastic zone the step size is determined using a non linear
equation. The proposed relation yields a large step size in
the elastic region and smaller and smaller sizes in the
plastic region. The mathematical representation of such
trend can be expressed in logarithmic profile as:
where us is the step radial displacement, n is the step
number, and N is the total number of steps.
Results and discussions
This section discusses the results on the deformation and
stress distributions along the roller expanded length and
transition zone in tube–tubesheet joint as well as the
effect of large clearance values on the interfacial stresses.
Figure 4 shows the tube inner and outer surface radial
deformations along the tubesheet width at the end of the
loading and unloading events. The radial displacement of
the inner tube surface is indicated by uri and that of the
outer surface by uro. Similar to what was reported in earlier work using 2-D axisymmetric model (Merah et al. 2010)
inner and outer radial displacements for 5% wall reduction
and an initial clearance of 0.127 mm remain constant in
most of the 47.25 mm expanded length. A disturbance in
the uri distribution happens in the transition zone (i.e. zone
between the expanded and unexpanded areas of the tube).
As will be explained later, this disturbance will have an
effect on the stress distributions. Releasing the load caused
the inner tube surface to spring back from a maximum of
0.232 to 0.188 mm resulting in radial strain of about 0.45%
and that of the outer surface from 0.190 to 0.151 mm.
To experimentally validate these results tube diame-
ters were measured before and after rolling. Several measurements were made for tube material having similar
value of tangent modulus, Ett, joint geometry and config-
uration, tube–tubesheet clearance, and loading condition
(i.e. Ett = 0.74 GPa, WR = 5% and c = 0.12 mm). A
micrometer was used to measure inner diameters of a
number of tubes before and after expansion to determine
the average value of the inner surface radial deformation.
The thick dashed line in Fig. 4 shows the average value of
the measured tube inner surface radial deformation
(0.185 mm). It is clear that the present 3-D FE model
yields an excellent representation of the deformed inner
surface.
Figure 5 illustrates the distributions of stress components
on the tube’s inner and outer surfaces along the axial
direction after the retraction of the rollers for the condition
of 5% WR and 0.127 mm radial clearance. The stresses
include rr, rh, ry, reqv, rcont which are the radial, tangential,
axial, equivalent and contact stresses, respectively. The
stresses corresponding to the tube’s outer surface display a
behavior similar to that of the radial displacement shown in
Fig. 4. These stresses remain uniform along most of the
expanded length and redistribute in the transition zone
between the fully expanded and the unexpanded tube. At the
end of the rolled tube length, the radial, axial and tangential
stress components (rr, rz, rh) exhibit two inflection points